Vol. 14, No. 1, 2021

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Rhombic planform nonlinear stability analysis of an ion-sputtering evolution equation

Sydney Schmidt, Stephanie Kolden, Bonni Dichone and David Wollkind

Vol. 14 (2021), No. 1, 119–142

A damped Kuramoto–Sivashinsky equation describing the deviation of an interface from its mean planar position during normal-incidence ion-sputtered erosion of a semiconductor or metallic solid surface is derived and the magnitude of the gradient in its source term is approximated so that it will be of a modified Swift–Hohenberg form. Next, one-dimensional longitudinal and two-dimensional rhombic planform nonlinear stability analyses of the zero deviation solution to this equation are performed, the former being a special case of the latter. The predicted theoretical morphological stability results of these analyses are then shown to be in very good qualitative and quantitative agreement with relevant experimental evidence involving the occurrence of smooth surfaces, ripples, checkerboard arrays of pits, and uniform distributions of islands or holes once the concept of lower- and higher-threshold rhombic patterns is introduced based on the mean interfacial position.

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ion-sputtered erosion, Kuramoto–Sivashinsky equation, Swift–Hohenberg equation, nonlinear stability analysis, rhombic pattern formation
Mathematical Subject Classification
Primary: 35B35, 35B36, 35R35, 74A50, 74K35
Received: 28 June 2020
Revised: 30 September 2020
Accepted: 10 October 2020
Published: 4 March 2021

Communicated by Martin J. Bohner
Sydney Schmidt
Department of Mathematics
Gonzaga University
Spokane, WA
United States
Stephanie Kolden
Department of Mathematics
Gonzaga University
Spokane, WA
United States
Bonni Dichone
Department of Mathematics
Gonzaga University
Spokane, WA
United States
David Wollkind
Department of Mathematics
Washington State University
Pullman, WA
United States